Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos

Abstract

The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, lambda(v), which is the growth or decay rate along rays at that velocity. We examine the behavior of.(v) for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by lambda((n) over cap upsilon(B) ((n) over cap)) = 0, with a generally direction-dependent butterfly speed v(B) ((n) over cap). In spatially local systems, lambda(v) is negative outside the light cone where it takes the form lambda(v) similar to -(v-v(B))(alpha) near v(B), with the exponent alpha taking on various values over the range of systems we examine. The regime inside the light cone with positive Lyapunov exponents may only exist for classical, semiclassical, or large-N systems, but not for “fully quantum” chaotic systems with strong short-range interactions and local Hilbert space dimensions of order one.